L(s) = 1 | − 2.32·2-s + 3.41·4-s − 5-s − 0.231·7-s − 3.29·8-s + 2.32·10-s + 3.13·11-s + 5.17·13-s + 0.539·14-s + 0.845·16-s + 3.12·17-s + 8.20·19-s − 3.41·20-s − 7.30·22-s + 2.60·23-s + 25-s − 12.0·26-s − 0.791·28-s + 1.75·29-s + 10.4·31-s + 4.63·32-s − 7.28·34-s + 0.231·35-s + 0.0476·37-s − 19.0·38-s + 3.29·40-s + 11.0·41-s + ⋯ |
L(s) = 1 | − 1.64·2-s + 1.70·4-s − 0.447·5-s − 0.0875·7-s − 1.16·8-s + 0.736·10-s + 0.946·11-s + 1.43·13-s + 0.144·14-s + 0.211·16-s + 0.758·17-s + 1.88·19-s − 0.764·20-s − 1.55·22-s + 0.542·23-s + 0.200·25-s − 2.36·26-s − 0.149·28-s + 0.325·29-s + 1.87·31-s + 0.818·32-s − 1.24·34-s + 0.0391·35-s + 0.00783·37-s − 3.09·38-s + 0.521·40-s + 1.73·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.082931544\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.082931544\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 + T \) |
good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 7 | \( 1 + 0.231T + 7T^{2} \) |
| 11 | \( 1 - 3.13T + 11T^{2} \) |
| 13 | \( 1 - 5.17T + 13T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 19 | \( 1 - 8.20T + 19T^{2} \) |
| 23 | \( 1 - 2.60T + 23T^{2} \) |
| 29 | \( 1 - 1.75T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 0.0476T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 5.76T + 43T^{2} \) |
| 47 | \( 1 - 1.91T + 47T^{2} \) |
| 53 | \( 1 - 2.56T + 53T^{2} \) |
| 59 | \( 1 - 0.116T + 59T^{2} \) |
| 61 | \( 1 + 0.140T + 61T^{2} \) |
| 67 | \( 1 + 0.235T + 67T^{2} \) |
| 71 | \( 1 - 7.10T + 71T^{2} \) |
| 73 | \( 1 + 8.81T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 8.88T + 83T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.510569384351468956890164732297, −7.891893583798151089250471881522, −7.25022597644804204516806297341, −6.51838100397637880956289232530, −5.82823243332567164684194398012, −4.62656072526418032218168208470, −3.57822804051385072071506544961, −2.82652706373219810003356124236, −1.29020890862314286542309330414, −0.947957122264042433291601338131,
0.947957122264042433291601338131, 1.29020890862314286542309330414, 2.82652706373219810003356124236, 3.57822804051385072071506544961, 4.62656072526418032218168208470, 5.82823243332567164684194398012, 6.51838100397637880956289232530, 7.25022597644804204516806297341, 7.891893583798151089250471881522, 8.510569384351468956890164732297